A174466 - OEIS

%I #26 Sep 08 2022 08:45:51

%S 1,7,10,31,16,70,22,111,64,112,34,310,40,154,160,351,52,448,58,496,

%T 220,238,70,1110,166,280,334,682,88,1120,94,1023,340,364,352,1984,112,

%U 406,400,1776,124,1540,130,1054,1024,490,142,3510,316,1162,520,1240

%N a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

%C Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.

%C tau(n) = A000005(n) = the number of divisors of n,

%C and sigma(n) = A000203(n) = sum of divisors of n.

%C Dirichlet convolution of A038040 and A000203. - _R. J. Mathar_, Feb 06 2011

%H Reinhard Zumkeller, <a href="/A174466/b174466.txt">Table of n, a(n) for n = 1..10000</a>

%F Logarithmic derivative of A174465.

%F Dirichlet g.f. zeta(s)*(zeta(s-1))^3. - _R. J. Mathar_, Feb 06 2011

%F a(n) = Sum_{d|n} tau_3(d)*d = Sum_{d|n} A007425(d)*d. - _Enrique Pérez Herrero_, Jan 17 2013

%F G.f.: Sum_{k>=1} k*tau_3(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Sep 06 2018

%F Sum_{k=1..n} a(k) ~ Pi^2*n^2/24 * (log(n)^2 + ((6*g - 1) + 12*z1/Pi^2) * log(n) + (1 - 6*g + 12*g^2 - 12*sg1)/2 + 6*((6*g - 1)*z1 + z2)/Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - _Vaclav Kotesovec_, Feb 02 2019

%o (PARI) {a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d,0))}

%o (Haskell)

%o a174466 n = sum $ zipWith3 (((*) .) . (*))

%o                   divs (map a000203 $ reverse divs) (map a000005 divs)

%o                   where divs = a027750_row n

%o -- _Reinhard Zumkeller_, Jan 21 2014

%o (Magma) [&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // _Marius A. Burtea_, Oct 18 2019

%Y Cf. A000005 (tau), A000203 (sigma), A007425 (tau_3), A034718, A038040, A174465.

%K nonn,mult

%O 1,2

%A _Paul D. Hanna_, Apr 04 2010